General solution to the Time-independent Schrödinger equation?

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Has anyone formulated a general solution to the time-independent Schrödinger equation in terms of the potential function V(r), and if so, what is it?

For any type of V(r).

So, instead of a differential equation, a direct relationship between the wavefunction and the potential.
 
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  • #3
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@BvU
Yes, I tried that first, but I couldn't find anything for, as mentioned, a general solution in terms of V(r).

And by general solution for V(r) I mean any type of V(r).

So I'm hoping some well-read physics experts here might know of it.
 
  • #4
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From the variety of approaches for different kinds of potential functions I estimate a zero probability for a (usable) general solution.
Would be something like a free lunch :cool:

some well-read physics experts
The more I think about it, the less I feel qualified :H
 
  • #5
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From the variety of approaches for different kinds of potential functions I estimate a zero probability for a (usable) general solution.
Would be something like a free lunch :cool:


The more I think about it, the less I feel qualified :H
A non-closed form solution would be really good too, hopefully there's one
 
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From what I've seen it's the other way around: suppose we have a ##\psi## that satisfies the TISE, what are its specific properties
 
  • #7
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From what I've seen it's the other way around: suppose we have a ##\psi## that satisfies the TISE, what are its specific properties
I believe the TISE generally produces families of discrete solutions, so a general solution would be in terms of both V(r) and parameters like quantum numbers, with the number of quantum numbers depending on the shape of V(r).
 
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we say the same thing
 
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